What integers can be written as the sum of two square numbers? Which ones can be written as the sum of three square numbers? Four? And for those numbers that can be written in such a way, how many ways can you do it?

These are questions that pop up in most courses in elementary number theory courses, and some results can be obtained using fairly elementary techniques. But the story does not end there, and these types of questions quickly lead into questions about topics such as Bernoulli numbers, L-functions, modular forms, and arithmetic progressions, many of which require quite a lot of sophisticated machinery to understand the definitions, let alone some of the deep and exciting results. The new book by Carlos J. Moreno and Samuel S. Wagstaff Jr, appropriately entitled Sums of Squares of Integers is designed to introduce the reader to this circle of ideas and the connections between the various topics. The prerequisites for the book are all topics that would be familiar to most beginning graduate students — some elementary number theory, some group theory, and complex analysis “at the level of Ahlfors.” Occassionally the authors will refer to more advanced topics, but for the most part the authors succeed in keeping the book quite self-contained.

Assume that a number of the form 2x2 can be written as the sum of two squares: that is, m2+ n2= 2x2. Then if follows that the numbers m2, x2, and n2 lie in an arithmetic progression. This observation connects the study of sums of squares of integers to the study of arithmetic progressions, and Moreno and Wagstaff use this connection to justify including a very nice chapter in their book on arithmetic progressions. This chapter includes a proof of the theorem of Szemeredi stating that every subset of the positive integers with positive asymptotic density contains arbitrarily long arithmetic progressions which has received quite a bit of attention recently due to the work of Green and Tao on prime numbers in arithmetic progressions.

The approach the authors take throughout their book is one that this reviewer would like to see in more books: they choose a relatively elementary set of questions and approach it from a variety of different angles — combinatorial, analytic, and algebraic — and in doing so they introduce topics from quite a few exciting areas of research and show some of the connections between these approaches. The authors end with a chapter about applications of these results to areas such as microwave radiation, diamond cutting, and cryptography. This is a nice touch, but to be honest I cannot imagine any reader reaching the end and not being convinced that these questions are inherently interesting and lead to some beautiful mathematics.

Elementary Methods
Introduction
Some Lemmas
Two Fundamental Identities
Euler's Recurrence for Sigma(n)
More Identities
Sums of Two Squares
Sums of Four Squares
Still More Identities
Sums of Three Squares
An Alternate Method
Sums of Polygonal Numbers
Exercises

Examples of Modular Forms
Introduction
An Example of Jacobi and Smith
An Example of Ramanujan and Mordell
An Example of Wilton: t (n) Modulo 23
An Example of Hamburger
Exercises

Hecke's Theory of Modular Forms
Introduction
Modular Group ? and its Subgroup ? 0 (N)
Fundamental Domains For ? and ? 0 (N)
Integral Modular Forms
Modular Forms of Type Mk(? 0(N);chi) and Euler-Poincare series
Hecke Operators
Dirichlet Series and Their Functional Equation
The Petersson Inner Product
The Method of Poincare Series
Fourier Coefficients of Poincare Series
A Classical Bound for the Ramanujan t function
The Eichler-Selberg Trace Formula
l-adic Representations and the Ramanujan Conjecture
Exercises

Representation of Numbers as Sums of Squares
Introduction
The Circle Method and Poincare Series
Explicit Formulas for the Singular Series
The Singular Series
Exact Formulas for the Number of Representations
Examples: Quadratic Forms and Sums of Squares
Liouville's Methods and Elliptic Modular Forms
Exercises